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August 6, 2008

This Week’s Finds in Mathematical Physics (Week 268)

Posted by John Baez

In week268 of This Week’s Finds
see a stunning view of Jupiter’s moon Io, and then learn about
Frobenius algebras in physics and logic…

… and a bit about
modular tensor categories and music theory, too!

Here you can see Io and Europa in front of Jupiter, sulfurous yellow and icy white:

A couple questions that came up:

Is the structure I described really equivalent to a **-autonomous category as
usually defined? I haven’t
had the energy to check.

What’s the relation between “cospans of finite sets as the free symmetric monoidal
category on a commutative separable Frobenius algebra” and “commutative separable Frobenius algebras and algebra morphisms as FinSetopFinSet^{op}”? It sounds like some Lawvere-ish thing where the category of free algebras is the opposite of the algebraic theory they’re algebras of… but here we’re dealing with PROPs, not algebraic theories! And cospans are getting into game…

Posted at August 6, 2008 7:12 PM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 268)

Hi John,

Here’s the precise reference to my paper with Dusko Pavlovic:

Bob Coecke and Dusko Pavlovic (2008) Quantum measurements without sums. In: The Mathematics of Quantum Computation and Technology, pp.559-596, eds. Chen, Kauffman and Lomonaco, Chapman and Hall/CRC. Also available as arXiv:quant-ph/0608035.

which is extranatural in the arguments a,ba, b (natural in aa and dinatural in bb), and which is compatible with the monoidal structure in a way which I’ll leave to your imagination (hint: θ−,b⊗c\theta_{-, b \otimes c} should be expressible in terms of components of θ−,b\theta_{-, b} and θ−,c\theta_{-, c}).

[Here’s one way of motivating that notion: pretend that we’re in the case where CC is symmetric monoidal closed. As you know, a strength on a functor F:C→CF: C \to C is equivalent to a structure of CC-enrichment on FF, in the sense of having a suitable family of natural maps

ab→FaFba^b \to F a^{F b}

where I am using exponential notation to denote the internal hom. To get the strength from the enrichment, just take the composite

a→(a⊗b)b→F(a⊗b)Fb,a \to (a \otimes b)^b \to F(a \otimes b)^{F b},

where the first map is coevaluation and the second uses enrichment, and use the ⊗−hom\otimes-\hom adjunction to transpose this to

a⊗Fb→F(a⊗b)a \otimes F b \to F(a \otimes b)

Similarly, given a CC-enriched contravariant functor F:C→CF: C \to C, we get composites

a→(a⊗b)b→FbF(a⊗b)a \to (a \otimes b)^b \to F b^{F(a \otimes b)}

which transposes to

F(a⊗b)⊗a→FbF(a \otimes b) \otimes a \to F b

as above in the case F=¬F = \neg.]

In order to have *-autonomy, you absolutely need to have such a strength on ¬\neg as part of the structure. [It gives you for instance “tertium non datur” a⊗¬a→¬Ia \otimes \neg a \to \neg I.] But I don’t see any way of getting that from your data.

In fact, without thinking about it too hard, it looks as though given any monoidal category CC and any contravariant adjoint equivalence ¬\neg on CC, you can rig a monoidal structure on CopC^{op} just by transporting the monoidal structure on CC across ¬\neg, so that ¬\neg is automatically monoidal and the adjoint equivalence lifts to a monoidal adjoint equivalence. But I’m feeling a little too lazy to check this out carefully just now.

Two final remarks:

The two notions of strength, covariant and contravariant, were discussed in a paper by Gerry Brady and myself on categorifying Peirce’s system Alpha [his existential graphs for propositional calculus]: we interpret his rule of iteration in categorified form as pertaining precisely to these notions of strength. We observed in our paper that the notion of *-autonomous category could be expressed in exactly these terms: a symmetric monoidal CC equipped with a strong contravariant adjoint equivalence ¬:C→C\neg: C \to C [where the unit and counit of the adjunction respect the strengths]. The paper appeared in JPAA [vol. 149 (3), 213-239].

IIRC, Robin Houston once explained to me a way of defining *-autonomous categories as commutative Frobenius pseudomonoids in the bicategory of categories and bimodules – perhaps he’d be willing to tell you more.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Sorry – you already did mention in TWF 268 that last point about *-autonomous categories being Frobenius pseudomonoids in the bicategory of categories and profunctors. I skipped over that by reading too quickly – my bad.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Todd wrote:

Sorry – you already did mention in TWF 268 that last point about *-autonomous categories being Frobenius pseudomonoids in the bicategory of categories and profunctors.

Yes, this is the result I was attempting to give people a feeling for. But since normal folks run away screaming when someone says “a *-autonomous category is just a Frobenius pseudomonoid
in the bicategory of categories and pseudofunctors”, I wanted to sneak up on this idea very, very cautiously.

So, I wanted to first sketch how *-autonomous categories naturally show up in logic, and then segue over to Street’s description of *-autonomous categories as Frobenius pseudomonoids.

But, I’m also trying to understand Melliès’ work on ‘game semantics’, so I wanted to work that in too…

I know that Melliès was very much influenced by your paper with Gerry Brady. So, he must know this result of yours, and he must have been trying to explain it to me:

… the notion of *-autonomous category could be expressed in exactly these terms:
a symmetric monoidal CC equipped with a strong contravariant adjoint equivalence ¬:C→C\not : C\to C [where the unit and counit of the adjunction respect the strengths].

But, the philosophy behind ‘game semantics’ seems to involve treating the two copies of CC as two ‘players’ — the guy who is fighting to prove some proposition, and the challenger who is trying to disprove them! The de Morgan duality built into logic is supposed to be nicely captured by this notion. To bring out this idea, Melliès prefers to talk about CC versus CopC^{op}, instead of two copies of CC.

This should be a minor esthetic decision, at least for defining **-autonomous categories… so I should be able to take your result and formulate it in terms of a covariant adjoint equivalence between CC and CopC^{op}. Right?

It’s supposed to be like defining a Frobenius algebra by saying you have an algebra AA, another algebra structure on A*A^*, and an isomorphism between them with some nice properties.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Well, there are various ways of expressing *-autonomy as you know. One which is more symmetric-looking, and close in spirit to the Frobenius pseudomonoid business, is that there is an adjunction which connects the tensor products:

a⊗b→ca→b*⊕c\frac{a \otimes b \to c}{a \to b^* \oplus c}

(where I am using ⊕\oplus to denote the dual tensor product, with slight misgivings for all the obvious reasons), and it is this interaction between the tensor products which one wants to express somehow. The suspicion I was expressing above is that with your setup, you’re not getting that level of interaction between the tensor products: unless I’m awfully mistaken [in which case I profusely apologize!], you’re getting basically just (categorified) De Morgan duality and double negation = identity, and that’s not enough.

Another way of putting it (and again I know you must know this) is that the structure must make provision for morphisms

a⊗a*→0a \otimes a^* \to 0

(where 00 denotes I*I^*) and

I→a*⊕aI \to a^* \oplus a

satisfying some trickily expressed triangular equations. The trickiness involves an interaction between the tensor products of the form

a⊗(b⊕c)→(a⊗b)⊕ca \otimes (b \oplus c) \to (a \otimes b) \oplus c

which simultaneously expresses a strength of −⊕c- \oplus c with respect to ⊗\otimes, and a co-strength of a⊗−a \otimes - with respect to ⊕\oplus. If you see how to get this from your setup or a modification of your setup, then I guess we’re both happy! :-)

Re: This Week’s Finds in Mathematical Physics (Week 268)

John wrote:

Melliès prefers to talk about CC versus CopC^{op}, instead of two copies of CC.

This should be a minor esthetic decision, at least for defining **-autonomous categories… so I should be able to take [Todd’s] result and formulate it in terms of a covariant adjoint equivalence between CC and CopC^{op}. Right?

Now I think the answer is yes — and I’ve attempted such a reformulation here.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Todd wrote:

In fact, without thinking about it too hard, it looks as though given any monoidal category CC and any contravariant adjoint equivalence ¬\not on CC, you can rig a monoidal structure on CopC^{op} just by transporting the monoidal structure on CC across ¬\not, so that ¬\not is automatically monoidal and the adjoint equivalence lifts to a monoidal adjoint equivalence. But I’m feeling a little too lazy to check this out carefully just now.

The only thing that could possibly go wrong is in lifting the adjoint equivalence to a monoidal adjoint equivalence. But this would be incredibly shocking, since I bet an adjoint equivalence is as close to an isomorphism of categories as we can get without going ‘evil’ and imposing equations between objects.

(Hmm, it would be fun to try to prove a precise theorem along these lines. Say everything you can say about the concept of ‘isomorphism of categories’ that’s expressible in the language of categories, functors, natural transformations and equations between natural transformations… not equations between functors or categories. I claim you’ve then given a definition of ‘adjoint equivalence’.)

Re: This Week’s Finds in Mathematical Physics (Week 268)

But this would be incredibly shocking

Sorry, I’m not sure whether you mean you believe me or don’t believe me! :-) Right now I’m leaning towards the former, but for what it’s worth, I overcame my earlier laziness and worked through the details, and I believe my earlier self! :-)

Re: This Week’s Finds in Mathematical Physics (Week 268)

Sorry for the ambiguity. I was trying to say it would be incredibly shocking if we could not lift an adjoint equivalence to a monoidal adjoint equivalence in the situation you described.

I tried to explain my intuition: obviously we can lift the adjoint equivalence if it’s an isomorphism, and “I bet an adjoint equivalence is as close to an isomorphism of categories as we can get without going ‘evil’ and imposing equations between objects.”

In other words: I bet that given any “non-evil” structure on a category CC, and an adjoint equivalence f:C→Df : C \to D, we can use ff to transfer this structure to DD and then lift ff to an adjoint equivalence respecting this extra structure.

And then I mused that it would be interesting to try to prove a precise theorem along these lines.

Doing this would require more formally understanding structures on categories that you can define using commutative diagrams that only impose equations at the 2-morphism — natural transformation — level. These are the ‘non-evil’ structures.

A typical example of a non-evil structure is ‘weak monoidal category’. A typical example of an evil one is ‘strict monoidal category’.

For any non-evil structure SS there should be a 2-category CatSCat_S of categories equipped with this structure, functors weakly but coherently preserving this structure, and natural transformations compatible with this structure.

We know how CatSCat_S is defined when SS is the structure ‘monoidal category’. Is there an automatic way to define CatSCat_S starting from SS? That’s the tricky part. But, I can imagine someone has already tackled it.

If this works, we get a forgetful 2-functor

F:CatS→CatF: Cat_S \to Cat

And then, we can try to prove that given a category C∈CatC \in Cat, and a way of equipping it with this extra structure, say C˜∈CatS\tilde{C} \in Cat_S with

F(C˜)=CF(\tilde{C}) = C

and an adjoint equivalence between CC and DD, we can lift this adjoint equivalence to one in CatSCat_S.

Clearly this is not the quick way to prove the result we’re talking about.

Re: This Week’s Finds in Mathematical Physics (Week 268)

That sounds a lot like the method in which a finitary algebraic theory, a first-order theory in which the axioms are all universal identities, like theories of groups or rings (but not fields), is converted into the category of all models of the theory and structure-preserving functions between them. This category of algebras has nice properties similar to the one you want: For example, every bijection between a group G and a set S induces a group structure on S, and the bijection lifts to a group isomorphism.

The right context for exploring this stuff is monads: every finitary algebraic theory induces a monad structure on Set, and the Eilenberg-Moore category is just the category of models. Is there already a categorified version of “monad” that seems plausibly relevant?

Re: This Week’s Finds in Mathematical Physics (Week 268)

In short: Frobenius algebras are lurking all over in physics, logic and quantum logic. There should be some unified explanation of what’s going on! Do you have any ideas?

I have tried to talk about this at various times here, but possibly never got close enough to the abstract truth to resonate with anyone else.

The latest refinement of what I got at for a long while was to relate it to weak resolutions of the point:

the walking Frobenius monad is weakly equivalent to the point

(if we disregard units at least, with them the statement is more subtle)

Here I mean the 2-category

- with a single object

- with 1-morphisms generated from a single endomorphism on that object

- with 2-morphisms generated from

a) triangles going from two copies of that 1-cell generator to one of them

b) and triangles going the other way round

- modulo the relations which make the triangles among themselves an associative product and coassociative coproduct and satisfy the Frobenius property with each other.

(So for the moment I am talking about a 2-category without unit 1-morphisms).

This 2-category is weakly equivalent to the point: the Frobenius property says precisely that all possible ways to go from a sequence of nn generating 1-cells to a sequence of mm generating 1-cells using the triangles and co-triangles are equal.

Back then, I tried to get at that using the notion of ambidextrous adjunctions: it seems there is a sense in which an ambidextrous adjunction is the “real” notion of weak equivalence: two objects related by an ambidextrous adjunction need not be equivalent. But still, somehow paths going back and forth between them run in a contractible space. Somehow. The monad generated from an ambidextrous adjunction is Frobenius.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Urs wrote:

Have to run now.

Sneaking out the door just when it gets tough?

Seriously: there are two famous ways for a 2-category to be ‘trivial’. One is being equivalent to the terminal 2-category. (By ‘equivalent’ I mean the sensible weak notion, not the silly strict one. Some people prefer the term ‘bi-equivalent’.) Another is having a contractible nerve.

The first property implies the second. The second property implies the first for 2-groupoids, but not general 2-categories.

As I sketched in week173, the walking adjoint equivalence is a 2-groupoid that’s contractible in both senses. I explained how how to see this geometrically: its nerve is homotopy equivalent to a 3-ball, and thus a point.

The walking equivalence is also a 2-groupoid, but it’s not equivalent to the terminal 2-category. We can see this geometrically too: its nerve is homotopy equivalent to a 2-sphere.

Maybe you are trying to say that while the walking ambidextrous adjunction is not equivalent to the terminal 2-category, its nerve is contractible.

That’s my best attempt to make sense of what you said!

I don’t know if it’s true, but it can be proved or disproved by drawing the nerve and seeing what it looks like.

But just before you ran away, you added an interesting extra twist: what nice condition on an ambidextrous adjunction guarantees that one or both of the resulting Frobenius monads are separable?

Perhaps we need to add such a condition to the walking ambidextrous adjunction to make its nerve contractible!

a) the Frobenius identity corresponds to the fusion move
for 2d triangulations

b) the separability / specialness to the bubble move

These two moves are sufficient to relate any two 2d triangulations rel
boundary. This is the reason why Frobenius algebras appear in state
sum models. Accordingly, they are sufficient to make any two parallel
2-morphisms in the walking Frobenius monad equal.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Is there a lesson to be learned from the difference between adjoint equivalences and equivalences? Unless I’m mistaken, the definition of equivalence is officially non-evil, but it’s still not the best thing. The weird thing for me is that I’ve always liked the concept of adjoint equivalence better than that of equivalence (not knowing till now that they weren’t equivalence in general 2-categories), but short of analyzing the homotopy type of their walking versions, I still can’t put my finger on why it’s better.

It would be great if the definition of equivalence violated a more subtle general rule than “don’t be evil”. Does it? I’m tempted to say the problem is with the implicit existential quantifier in the word “isomorphism”. But it seems easy enough to get rid of that by adding the inverses to the homomorphisms as part of the data, so I guess I’m stuck. Something like “only add one piece of data at a time, and when you do also be sure to X” is what I have in mind.

Re: This Week’s Finds in Mathematical Physics (Week 268)

James wrote:

Is there a lesson to be learned from the difference between adjoint equivalences and equivalences? Unless I’m mistaken, the definition of equivalence is officially non-evil, but it’s still not the best thing. The weird thing for me is that I’ve always liked the concept of adjoint equivalence better than that of equivalence […] but short of analyzing the homotopy type of their walking versions, I still can’t put my finger on why it’s better.

We don’t really need to take homotopy types of nerves to say what’s going on here. We can say it a different way — which may or may not make you feel more enlightened.

Namely: there’s a little category called ‘the walking isomorphism’: two objects and an isomorphism between them. This category is equivalent to ‘the walking object’: the category with just one object and its identity morphism.

This is a powerfully precise way of saying that “having two things with an isomorphism between them is just like having one thing”.

To see this, let 11 be the walking object and let IsoIso be the walking isomorphism. Then our equivalence

1≃Iso1 \simeq Iso

automatically gives an equivalence

hom(1,C)≃hom(Iso,C) hom(1,C) \simeq hom(Iso,C)

for any category CC. Of course

hom(1,C)≃Chom(1,C) \simeq C

On the other hand, hom(Iso,C)hom(Iso,C) is the category of ‘pairs of objects in CC equipped with an isomorphism between them’.

So, the category of ‘pairs of objects in CC equipped with an isomorphism between them’ is equivalent to CC.

This nicely formalizes the idea that having two things with an isomorphism between them is just like having one thing!

On the other hand, look what happens when we go up to 2-categories.

Let 11 be the 2-category with one object, its identity morphism, and its identity 2-morphism. And let EquivEquiv be the walking equivalence: the 2-category consisting of two objects and an equivalence between them.

We might think that 11 was equivalent to EquivEquiv, but it’s not true!

So, when we’re in a 2-category — e.g. Cat — having two things with an equivalence between them is not just like having one thing.

On the other hand, let AdEquivAdEquiv be the ‘walking adjoint equivalence’. Then we have

1≃AdEquiv 1 \simeq AdEquiv

so for any 2-category

C≃hom(1,C)≃hom(AdEquiv,C) C \simeq hom(1,C) \simeq hom(AdEquiv,C)

So: having two things and an adjoint equivalence between them is just like having one thing.

Technical note: here I’m using the sensibly weakened concept of equivalence between 2-categories, which some people call ‘biequivalence’.

Puzzle: I said that the equivalence

C≃hom(Iso,C)C \simeq hom(Iso,C)

meant having one object in CC was just like having two objects in CC and an isomorphism between them. But later I said the concept of equivalence was in some sense defective: having two things and an equivalence between them is not just like having one thing!

So: is having CC and hom(Iso,C)hom(Iso,C) just like having CC, or not?

Re: This Week’s Finds in Mathematical Physics (Week 268)

Thanks. That’s interesting, but I guess I hoping for something closer to the syntactic level, something along the lines of “First make all existential quantifiers part of the structure, and then if you can state a relation in terms of your structure, you should”.

For example, what does it mean for CC to be equivalent to DD? One definition is that we have two functors f,gf,g such that fg≅1Cfg\cong 1_C and gf≅1Dgf\cong 1_D. But there are ∃\existss hiding in the ≅\congs, so we should really add structure. Then we get the data of a map α:fg→1C\alpha:fg\to 1_C, a map β:gf→1D\beta:gf\to 1_D, and their inverses α−1\alpha^{-1} and β−1\beta^{-1}. So now we’ve converted what used to be properties on ff and gg into structure. But now that we have that structure, it’s possible to state the triangle axioms, so (by the principle above) we should! Then we have the concept of adjoint equivalence.

Is this principle a reasonable way of understanding why equivalence is not as good as adjoint equivalence? (It seems very close to the requirement that all homotopies be contractible.) If so, what is its most natural general formulation?

Re: This Week’s Finds in Mathematical Physics (Week 268)

James wrote:

That’s interesting, but I guess I hoping for something closer to the syntactic level, something along the lines of “First make all existential quantifiers part of the structure, and then if you can state a relation in terms of your structure, you should”.

Okay. This motto seems to consist of two separate parts. I was focusing on this part:

“…then if you can state a relation in terms of your structure, you should.”

In topology this shows up in the recipe for ‘killing homotopy groups’, which says: “whenever you see a map from SjS^j into your space XX, use this to glue on an (j+1)(j+1)-disk.” This makes the jjth homotopy group of your space trivial. It creates an enormous (j+1)(j+1)st homotopy group, which you can then go ahead and kill by the same method, if you want.

Translated into nn-category language, this recipe says “whenever you see two jj-morphisms f,g:x→yf,g: x \to y, throw in an (j+1)(j+1)-morphism α:f→g\alpha: f \to g.” We can do this starting with whatever jj we want and working our way up. When we’re working with nn-categories, and jj reaches nn, a (j+1)(j+1)-morphism is the same as an equation, and we can stop there.

(All equations between two fixed nn-morphisms are automatically equal — so all the higher-dimensional holes are filled in by definition.)

All this sounds a bit complicated, but it’s actually a way of making an nn-categorical structure as boring as possible. We do it when we don’t want complications: when we want a bunch of things to be equivalent whenever possible. This is the sense in which an adjoint equivalence is less complicated than an equivalence.

The other part of your motto:

“First make all existential quantifiers part of the structure…”

is another important aspect of the category-theorists’ worldview, but perhaps logically separate. Simply put: knowing that something exists is not nearly as useful as having it in your hands.

Somehow combining these mottos can lead us to the definition of adjoint equivalence.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Good. Thanks. I learned about the ∃\exists-removing principle a long time ago but didn’t know about the state-whatever-axioms-as-you-can principle. It’s tempting to call old-fashioned equivalence an under-achiever, what with it just sitting there leaving extra axioms unstated. I’m also tempted to say that “equivalence” should be redefined to be adjoint equivalence. This shouldn’t be a problem, because they were the same anyway in CAT. Maybe 2-category-theorists would object, but hey, they probably still call themselves bicategory-theorists.

By the way, do you know when this distinction between equivalence and adjoint equivalence was first noticed?

Regarding the puzzle, I’m not sure I understand the question. Are you asking whether CC is adjoint equivalent to hom(Iso,C)\mathrm{hom}(\mathrm{Iso},C)?

Re: This Week’s Finds in Mathematical Physics (Week 268)

I never quite got to the main point when explaining the work of Coecke, Pavlovic and Vicary. Maybe this addition will help:

So, the category with:

commutative separable complex Frobenius algebras as objects;

algebra homomorphisms as morphisms

is equivalent to FinSetopFinSet^{op}. This means we can find the category
of finite sets — or at least its opposite, which is just as good —
lurking inside the world of Frobenius algebras!

Coecke, Pavlovic and Vicary explore the ramifications of this result
for quantum mechanics, using Frobenius algebras that are Hilbert
spaces instead of mere vector spaces. Taking a Hilbert space and
making it into a commutative separable Frobenius algebra is the same making it into a commutative separable Frobenius algebra is the same
as equipping it with an orthonormal basis. There’s no general
way to duplicate quantum states - “you can’t clone a
quantum” - but if you only want to duplicate states lying in a
chosen orthonormal basis you can do it. So, you can think of
commutative separable Frobenius algebras as “classical data
types”, which let you duplicate information. The result I just
sketched shows that such Frobenius algebras are secretly just like
finite sets. So, we now see how to describe finite sets starting from
Hilbert spaces and introducing a notion of “classical data
type” formulated purely in terms of quantum concepts! - that is,
linear algebra.

Re: This Week’s Finds in Mathematical Physics (Week 268)

John, thanks for explaining all this so carefully. So indeed, in FdHilb orthonormal basis and commutative dagger Frobenius algebras are exactly the same thing, and the latter captures classicality in terms of actual capabilities (copying/deleting) while the first just gives us the set of classical data in a somewhat boring manner merely by providing elements.

But really interesting things happen if you look for these commutative dagger Frobenius algebras in other categories that one might want to use for modeling certain fragments of quantum theory.

In Rel, even on the 2-element set, there are already two “incompatible” and even “complementary” commutative dagger Frobenius algebras! (typical examples of complementarity are position/momentum or spin Z/spin X) These are:

δg = 0 ~ (0,0); 1~(1,1)
εg = 0 ~ *; 1 ~ *

δr = 0 ~ {(0,0),(1,1)}; 1~{(0,1),(1,0)}
εr = 0 ~ *

The first one is the one which one expects but the second one is a somewhat mysterious creature.

Now, recall that in TWF251 John discussed Rob Spekkens’ toy model and mentioned that it would be good to have a categorical understanding of that. It turns out that the way to understand this model is exactly in terms of commutative dagger Frobenius algebras. What corresponds to bases in his model are a pair of the above mentioned mysterious (δr, εr) on the four element set. There are three ways of partitioning a four elemement set in two giving rise to the X, Y and Z directions.

A paper by my student Bill Edwards and I on this should be at the arXiv tomorrow but here’s already a version of Toy quantum categories.

Bob’s paper

That’s a very interesting paper, Bob. We have been looking at the relation between mutually unbiased bases and braid group representations using finite fields. The p=3 case is associated to mass quantum numbers, in analogy to spin for p=2, although in order to understand mass one does need to consider a lot more structure (monoidality has to go, for starters).

Re: This Week’s Finds in Mathematical Physics (Week 268)

So indeed, in FdHilb orthonormal basis and commutative dagger Frobenius algebras are exactly the same thing

Jamie and I have discussed this before after he explained it to me, so perhaps he can correct me if I am confused, but don’t you really mean separable commutative dagger Frobenius algebras? My understanding is that to specify a commutative daggger Frobenius algebra, one needs to specify the idempotents eie_i and the scale factors λi=g(ei,ei)\lambda_i = g(e_i, e_i) (if the algebra is separable, these factors are unity). So the information contained in a general commutative dagger Frobenius algebra is not just an orthonormal basis eie_i, but a weighted orthonormal basis. Those weightings are precisely the things which give us the TQFT invariants for a genus g surface:

(1)Z(Σg)=∑iλig−1.
Z(\Sigma_g) = \sum_i \lambda_i^{g-1}.

Perhaps I am not on the same boat as far as what the morphisms are required to be: I am working in the paradigm where a morphism of commutative dagger Frobenius algebras is defined as a morphism of Frobenius algebras (i.e. not just a morphism of algebras).

Re: This Week’s Finds in Mathematical Physics (Week 268)

You indeed want the comultiplication to be isometric i.e. δ† o δ - or ‘special’ as it is sometimes called. I tend to forget to mention that since in QM terms this merely means physical realisability - if one believes ‘unitary = realisable’.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Bob wrote:

I tend to forget to mention that since in QM terms this merely means physical realisability - if one believes ‘unitary = realisable’

Well this increases the confusion even more… because a commutative dagger Frobenius algebra is exactly the image of the circle under a unitary 2d TQFT, and hence certainly realizable as a genuine ‘physical model’. However I guess you are talking about ‘unitary’ and ‘realizable’ in quantum computing terms. Mmm… it might be a bit confusing if different parts of the quantum world use the word unitary in different ways. In the TQFT paradigm, unitary simply means ‘preserves duals’ (as opposed to ‘every morphism gets sent to a unitary operator’) so the equation δ†∘δ=1\delta^\dagger \circ \delta = 1 is not required since it is not part of that requirement (nor does it hold in 2Cob).

John wrote:

Do you know the cute string diagram proof that a morphism of Frobenius algebras (i.e. not just their underlying algebras) is automatically an isomorphism?

No I didn’t before you asked me… but is it the one where you define A−1A^{-1} by ‘bending’ the cylinder round, and then prove A−1∘A=1A^{-1} \circ A = 1 by ‘squeezing’ the A’s off the cylinder like toothpaste? :-) Yeps that is pretty cool, thanks for alerting me to this!

Eric wrote:

I’m not sure if this is relevant, but given a complete set of idempotents, we can define a nice abstract graded differential algebra on them. From there, we could construct a graph…

Yes it is all relevant. One day it will all form part of a nice coherent picture… but as far as my expertise is concerned at this stage I can only guess the following concepts will/have come into the mix: quivers, coherent sheaves, cohomology, sigma model, fermions, Poincare duality. It is entirely possible that Kevin Costello has already attained Shangri-la on these issues :-)

Re: This Week’s Finds in Mathematical Physics (Week 268)

Bruce wrote:

Perhaps I am not on the same boat as far as what the morphisms are required to be: I am working in the paradigm where a morphism of commutative dagger Frobenius algebras is defined as a morphism of Frobenius algebras (i.e. not just a morphism of algebras).

Do you know the cute string diagram proof that a morphism of Frobenius algebras (i.e. not just their underlying algebras) is automatically an isomorphism?

I ask because I only learned it recently, and found it surprising at first. That’s why I mentioned it in week268.

So, your paradigm gives a groupoid of commutative dagger Frobenius algebras. To get the category of finite sets starting from Frobenius algebras, one needs another paradigm.

Frobenius algebras are defined by a PROP that happens to be compact (viewed as a symmetric monoidal category). I hear that for any sort of algebraic gadget defined by a PROP that is compact, the category of these gadgets is actually a groupoid. I haven’t checked this yet.

Re: This Week’s Finds in Mathematical Physics (Week 268)

I hear that for any sort of algebraic gadget defined by a PROP that is compact, the category of these gadgets is actually a groupoid. I haven’t checked this yet.

This sounds familiar. It reminds me of a general result: that the restriction of a lax transformation between homomorphisms of bicategories F,G:B→CF, G: B \to C, to the (locally full) sub-bicategory Ladj(B)Ladj(B) whose 1-cells are left adjoints in BB, is a strong transformation. In fact, I’m thinking that the result stated above is a special case of this general result.

Thinking out loud here: we are considering monoidal categories as one-object bicategories, strong monoidal functors as homomorphisms, and monoidal transformations as lax transformations. To say that a monoidal category BB is compact is to say that Ladj(B)=BLadj(B) = B on the nose.

Okay, now that I’ve convinced myself with these mutterings, I might as well run through it in this special case. So the claim is that if BB is any compact monoidal category (not just a PROP) and CC is monoidal, and F,G:B→CF, G: B \to C are (strong) monoidal functors, and θ:F→G\theta: F \to G is a monoidal transformation, then the component θ(b)\theta(b) is invertible at each object bb of BB, so that θ\theta is an invertible monoidal transformation.

Indeed, the inverse to θ(b):Fb→Gb\theta(b): F b \to G b is constructed in just the way you’d expect: as the composite

where η\eta and ε\varepsilon are the evident unit and counit. An easy string diagram argument, parallel to the neat thing you pointed out for morphisms of Frobenius algebras, shows that this is indeed the inverse.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Bruce said:

My understanding is that to specify a commutative daggger Frobenius algebra, one needs to specify the idempotents eie_i and the scale factors λi=g(ei,ei)\lambda_ i=g(e_i,e_i) (if the algebra is separable, these factors are unity)

This isn’t quite true — for the Frobenius algebra to be separable, the complex numbers λi\lambda_i only need to have unit magnitude. They don’t have to be real.

He also said:

Those weightings are precisely the things which give us the TQFT invariants for a genus g surface:

(1)Z(Σg)=Σiλig−1
Z ( \Sigma _g ) = \Sigma_i \lambda_i ^{g-1}

I think we actually end up summing |λi|2(g−1)|\lambda_i| ^{2(g-1)}. After all, the handle operator multiplies the iith subspace by |λi|2|\lambda_i| ^2. We could replace each λi\lambda_i with its magnitude |λi||\lambda_i| and we’d get an isomorphic TQFT, but the TQFTs themselves have this extra degree of freedom. This is why I keep going on about complex numbers rather than just real weightings!

Re: This Week’s Finds in Mathematical Physics (Week 268)

Bleg: I vaguely recall a Seiberg-Moore paper that described a 2D TQFT that had various 6j-type things associated to critical points of surfaces from the early 1990s, but I can’t seem to find the article on the arXiv. Does anyone else remember this?

In regard to Frobenius algebras, our paper CCEKS will appear in the Lin memorial issue. I’ll update the arXiv version soon (stupid typos in the example calculations). These ideas seem to work for Frobenius algebra objects in any category. I am not sure to what extent the 2 and 3-cocycles help define 2- or 3- categorical versions thereof. My head gets too fuzzy when I think about these things.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Scott wrote:

Bleg: I vaguely recall a Seiberg-Moore paper that described a 2D TQFT that had various 6j-type things associated to critical points of surfaces from the early 1990s, but I canít seem to find the article on the arXiv.

Are you talking about the super-famous paper by Moore and Seiberg which got solutions of the pentagon and hexagon equations from rational conformal field theories? That’s:

I think it’s a bit confusing using this adjective ‘separable’ to mean one thing for Frobenius algebras, that comultiplication followed by multiplication gives the identity, and something else for mere algebras, where it’s a property of semisimple algebras. What’s silly about this is that it’s quite possible to give a separable algebra a Frobenius form which turns it into a non-separable Frobenius algebra.

But I guess it must be true that every separable Frobenius algebra has a separable underlying algebra, which would explain the terminology, and your statement about commutative separable Frobenius algebras. Can someone prove this to me? I’d be happy with a proof of this special case. Also, this leads me to suspect that a commutative algebra can be made into a separable Frobenius algebra in at most one way — does anybody know about this?

You say that “taking a Hilbert space and making it into a commutative separable Frobenius algebra is the same as equipping it with an orthonormal basis”, but I don’t think this is quite true. In fact, I can prove it’s not true — the inner product of the Hilbert space plays no part in the definition of a commutative separable Frobenius algebra, so if you found an example where what you’re saying is true, you could just choose a different inner product under which the basis isn’t orthonormal any more.

I think the correct result is that, on a finite-dimensional complex vector space,
choosing a commutative separable Frobenius algebra is equivalent to choosing a basis. There’s no ‘orthonormal’ any more, because that doesn’t mean anything for a mere vector space. To get the basis from the commutative separable Frobenius algebra, look at all the vectors that are perfectly copied by the comultiplication. To get the commutative separable Frobenius algebra from the basis, choose the unique one that has a comultiplication which copies your chosen basis elements!

Given a basis for a complex vector space, there’s a unique inner product that makes that basis orthonormal. Let’s say we choose that inner product for the vector space that has our commutative separable Frobenius algebra on it. How will we know when we’ve chosen the correct inner product? Because this is the unique inner product for which the Frobenius comultiplication is adjoint to the algebra multiplication! We’re allowed to talk about adjoints because we’ve chosen an inner product, and since we’re in a finite-dimensional space it’s guaranteed that adjoints will exist.

Given this, it should be pretty believable that, as Bob said above in different words, for a finite-dimensional Hilbert space, choosing a commutative separable Frobenius algebra, for which the multiplication is adjoint to the comultiplication, is equivalent to choosing an orthonormal basis.

So far, the star of the show has been this separability property of our commutative Frobenius algebras, which seems to be doing the impressive job of ensuring that the underlying algebra is semisimple. Bob Coecke, Dusko Pavlovic and I recently found that there’s another neat way to ensure this: for a Frobenius algebra on a Hilbert space, require that the comultiplication is adjoint to the multiplication! We call these ‘†\dagger-Frobenius algebras’, because the adjoint to a linear map f:A→Bf:A \to B is often written f†:B→Af ^\dagger : B \to A. The Frobenius algebra doesn’t have to be commutative for this to work, and it doesn’t have to be separable. (Of course, the underlying algebra is separable. This is why it’s confusing using ‘separable’ as an adjective for Frobenius algebras!) However, for the ‘daggerdagger’ to make sense, it has to be a Frobenius algebra on a Hilbert space, not just a vector space. So, the motto is †\dagger-Frobenius algebras are semisimple.

But, what if we had Hilbert space HH equipped with a †\dagger-Frobenius algebra that is commutative? Then we’d have a commutative semisimple algebra on the Hilbert space. One way to get a basis out of this is to find all the vectors ϕ:ℂ→H\phi: \mathbb{C} \to H which are copied by the comultiplication m†:H→H⊗Hm ^\dagger : H \to H \otimes H, satisfying m†∘ϕ=ϕ⊗ϕm ^\dagger \circ \phi = \phi \otimes \phi. These define a basis, and there’s a fun picture-based proof that this basis is orthogonal. Have a go! (There’s an easy proof in the case that our †\dagger-Frobenius algebra is separable, but we’re not assuming that here.)

However, these basis elements won’t be normalised in general. So, for a finite-dimensional Hilbert space, choosing a commutative †\dagger-Frobenius algebra, for which the multiplication is adjoint to the comultiplication, is equivalent to choosing an orthogonal basis. The basis elements will be normalised exactly when the commutative †\dagger-Frobenius algebra is separable.

I think the proof breaks down into steps like this. It’s really a fact about algebras, not Frobenius algebras.

First, assume you have an associative algebra over any field of characteristic zero. Then this algebra is strongly separable iff it is finite-dimensional and semisimple — see the
paper by Aguiar that I cited in week268. Recall that an algebra
is strongly separable iff the bilinear form
g(a,b)=tr(LaLb)g(a,b) = tr(L_a L_b)
is nondegenerate. It’s semisimple iff has no nontrivial nilpotent ideals.

Next, use the version of the Artin–Wedderburn theorem that says any finite-dimensional semisimple algebra over the complex numbers is a finite direct sum of complex matrix algebras.

(Alas, all the statements of the Artin–Wedderburn theorem that I can find online are so general that it takes real work to see how this special case follows. If you can grab a decent intro text on algebra, it should have this result, which is sometimes just called ‘Wedderburn’s theorem’ — Artin is the guy to blame for generalizing it almost beyond recognition. But beware, there’s another theorem called Wedderburn’s theorem, about finite division rings.)

Then, notice that a finite direct sum of complex matrix algebras is commutative iff it is a finite direct sum of copies of ℂ\mathbb{C}.

I would appreciate more comments from algebraists about this circle of results!

I think it’s a bit confusing using this adjective ‘separable’ to mean one thing for Frobenius algebras, that comultiplication followed by multiplication gives the identity, and something else for mere algebras, where it’s a property of semisimple algebras.

I agree. And I’m afraid I may have started this annoying usage.

I wanted to use ‘special’, the way you do — but then I saw that Schweigert et al use that to mean something slightly different, which agrees in VectVect (I think) but not all monoidal categories!

What’s silly about this is that it’s quite possible to give a separable algebra a Frobenius form which turns it into a non-separable Frobenius algebra.

Right. To add to the confusion, the relevant property for algebras is actually called ‘strongly separable’, as defined above — for most algebraists, ‘separable’ means something else, as explained in Aguiar’s paper!

R. Rosebrugh, N. Sabadini and R.F.C. Walters seem to use use ‘commutative separable algebra’ to mean a commutative Frobenius monoid in any symmetric monoidal category. This was my excuse for speaking of ‘separable Frobenius algebras’.

To show just how little agreement there is about these terms, Wikipedia defined ‘Frobenius algebra’ as an algebra with the property that it admits a nondegenerate bilinear form with g(ab,c)=g(a,bc)g(a b,c) = g(a,b c), instead of an algebra equipped with such a bilinear form. At least, they did before I changed it!

So, maybe we just need to step in and clear things up, with proper respect for tradition but also a certain boldness. People studying ordinary algebras have their own terminology which we should interfere with — like ‘separable’ and ‘strongly separable’. But the terminology for Frobenius algebras may require a bit of reform. Maybe I should start by rewriting week268 to call Frobenius algebras with this property:

‘special’ instead of ‘separable’.

But maybe we should read the Frobenius algebra literature more deeply before jumping in and messing things up more.

Re: This Week’s Finds in Mathematical Physics (Week 268)

John said:

You say that “every commutative separable Frobenius algebra over the complex numbers looks like ℂ⊕⋯⊕ℂ”. I didn’t know this!

[…] It’s really a fact about algebras, not Frobenius algebras.

I’m really confused now! The result we’re talking about definitely has to do with Frobenius algebras, inasmuch as it’s got the word “Frobenius” in it. I suppose you mean that the proof relies on a fact about plain old algebras — but we’re going to have to connect this fact to Frobenius algebras somehow, which surely won’t be trivial.

Now I’m worried that when you wrote “commutative separable Frobenius algebra”, the “separable” wasn’t being used in its comultiplication-followed-by-multiplication-gives-the-identity meaning! Can you confirm which meaning you meant?

Recall that an algebra
is strongly separable iff the bilinear form
g(a,b)=tr(LaLb)g(a,b)=tr(L _a L_ b)
is nondegenerate.

OK. Playing with some pictures, I reckon that if a Frobenius algebra has comultiplication followed by multiplication giving the identity, then tr(LaLb)tr(L_a L_b) will indeed be nondegenerate. It confuses me that you don’t explicitly say this… surely it’s a necessary part of the proof!

I wanted to use ‘special’, the way you do — but then I saw that Schweigert et al use that to mean something slightly different, which agrees in Vect (I think) but not all monoidal categories!

I suppose you’re talking about their definition on page 23. They define “special” to mean that comultiplication followed by multiplication is proportional to the identity — but surely that’s not going to be equivalent to what we’re calling special, even in Vect.

They also have this weird extra condition that unit followed by counit equals cup followed by cap. What I don’t like about this is that it seems to rely on the fact that, for each self-dual object, cup followed by cap is well-defined — I don’t know a counterexample, but it would amaze me if this were true in all symmetric monoidal categories. They’re working in modular tensor categories, where you can presumably rely on things like this. Even more confusingly, this condition is redundant, anyway! If cup followed by cap is well-defined, and we have a Frobenius algebra with comultiplication followed by multiplication proportional to the identity, then we’re guaranteed that unit followed by counit is proportional to cup followed by cap, by the same constant of proportionality.

So, I think it’s safe to forget about this extra condition. What we’re calling “special” is what they call “normalised special” — but then they say that they’re just going to call this “special” anyway, because they’re not interested in the non-normalised case.

R. Rosebrugh, N. Sabadini and R.F.C. Walters seem to use use ‘commutative separable algebra’ to mean a commutative Frobenius monoid in any symmetric monoidal category.

Aha — they’re using “commutative separable” to mean “commutative Frobenius, and comultiplication followed by multiplication gives the identity”. On the face of it this seems pretty egregious, seeing that “separable algebra” already means something else. But given what we are talking about above, I reckon it’s true that a commutative algebra which is separable in the traditional sense admits a unique special Frobenius structure, as long we we stick to finite-dimensional complex algebras! So their definition is actually inspired. If this is right, though, then it’s a bit weird that they don’t explain this anywhere.

Maybe I should start by rewriting week268 to call Frobenius algebras with this property … ‘special’ instead of ‘separable’.

Re: This Week’s Finds in Mathematical Physics (Week 268)

We’re talking about the fact that every commutative special Frobenius algebra over the complex numbers is a direct sum of copies of ℂ\mathbb{C}.

John wrote:

It’s really a fact about algebras, not Frobenius algebras.

Jamie wrote:

I’m really confused now! The result we’re talking about definitely has to do with Frobenius algebras, inasmuch as it’s got the word “Frobenius” in it.

Sure — but there’s no real difference between what you call a ‘special Frobenius algebra’ and what algebraists call a ‘strongly separable algebra’. Each one can be made into the other in a canonical way.

Playing with some pictures, I reckon that if a Frobenius algebra has comultiplication followed by multiplication giving the identity, then tr(LaLb)tr(L_a L_b) will indeed be nondegenerate. It confuses me that you don’t explicitly say this… surely it’s a necessary part of the proof!

It’s necesssary, but I figured this part would be easy for you — and it was.

Anyway, let’s be really precise and put an end to any lingering confusion.

A finite-dimensional associative algebra has the property that tr(LaLb)tr(L_a L_b) is nondegenerate if and only if there exists a way to give it a comultiplication and counit making it into a Frobenius algebra such that comultiplication followed by multiplication is the identity. And, if such a way exists, it’s unique.

So, any special Frobenius algebra (in your sense) has an underlying algebra that is strongly separable. And conversely, any strongly separable algebra can be extended to a special Frobenius algebra in a unique way.

So, classifying special Frobenius algebras is just the same as classifying strongly separable algebras.

Re: This Week’s Finds in Mathematical Physics (Week 268)

John said:

A finite-dimensional associative algebra has the property that tr(LaLb)\mathrm{tr}(L _a L _b) is nondegenerate if and only if there exists a way to give it a comultiplication and counit making it into a Frobenius algebra such that comultiplication followed by multiplication is the identity. And, if such a way exists, it’s unique.

That’s what I suspected, thanks for spelling it out.

It seems to me that for a special Frobenius algebra we have tr(LaLb)=g(a,b)\mathrm{tr}(L_a L_b) = g(a,b), where gg is the Frobenius form. Since gg is necessarily nondegenerate, this explains why tr(LaLb)\mathrm{tr}(L_a L_b) is nondegenerate. A surprising corollary of this is that g(a,b)=tr(LaLb)=tr(LbLa)=g(b,a)g(a,b) = \mathrm{tr}(L_a L_b) = \mathrm{tr}(L_b L_a) = g(b,a) — and so every special Frobenius algebra is also symmetric! This would work for Frobenius algebras on any object in a monoidal category with a cyclic trace.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Jamie wrote:

It seems to me that for a special Frobenius algebra we have tr(LaLb)=g(a,b)tr(L_a L_b)=g(a,b), where gg is the Frobenius form.

Right — and this is precisely where playing around with pictures is so helpful. Drawing tr(LaLb)=g(a,b)tr(L_a L_b)=g(a,b) as a picture makes everything clear! The Wizard drew this picture on February 26, 2001:

This is just the definition of ‘special’ with the output pulled up to become an input! The Wiz used this as part of his proof that we get 2d topological lattice field theories from finite-dimensional semisimple algebras.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Just wanted to point to another reference where some of the facts you have been
discussing have been collected: math.QA/0602047. Note that this paper is often using results from Aguiar’s paper.

Theorem 2.6 recalls the fact that A finite dimensional over *any* field k, with canonical bilinear form nondegenerate is equivalent to A being strongly separable. Note that every strongly separable algebra is then a symmetric Frobenius algebra.

Then the following facts are recalled:

Let A be an algebra over some field k.
If A is strongly separable, then A is finite-dimensional, separable, and semisimple.
If A is separable and commutative, then A is strongly separable. If A
is finite-dimensional and semisimple and char k = 0, then A is strongly separable. If A is finite-dimensional and semisimple and k is a perfect field, then A is separable.

Note that for k a field and Mn(k) the algebra of (n x n)-matrices over k, if
char k divides n, then Mn(k) is semisimple and separable, but not strongly
separable.

In the context of a generic symmetric monoidal category it is shown (Proposition
2.13) that the canonical pairing is given by multiplication by the “window element”, sometimes called the “handle element”. The window element is a central
invertible element of Hom(1,A). Theorem 2.14 shows that in a general symmetric
monoidal category, the symmetric Frobenius algebra A is strongly separable iff
the window element is invertible. With the definition of special that you are
using above, your window element would be 1.

Proposition 2.15 shows that for A a symmetric Frobenius algebra in any symmetric
monoidal category such that dim A is invertible in Hom(1,1), then special in the
sense of Runkel, Fjelstad, Fuchs, and Schweigert is equivalent to strong
separability and the condition that the window element is some invertible element
of Hom(1,1).

One important fact to mention is that the “special” condition is related to a
very, um special, case of strongly separablity. In general, the window element of
a strongly separable algebra is just a central invertible element of Hom(1,A).
The “special” condition forces this element to be of the form multiplication by an element in Hom(1,1). Is it possible that this is the difference between the notion of a ‘special Frobenius algebra’ and what algebraists call a ‘strongly separable algebra’.

I know that the conversation has been focused on the case when k is the field of
complex numbers. However, in the state sum open-closed TQFTs that Hendryk
Pfeiffer and I studied, having the window element = 1 is exactly the right
condition to impose in order to have the open sector of the TQFT be trivial. The
window element plays a crucial role. Furthermore, in the applications to knot
theory math.GT/0606331 that Pfeiffer and I studied, we found that working over a field of finite characteristic was a valuable tool. In that case, one needs to be very careful
about what terminology one is using.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Hi, Aaron! Yes, I gave links to your papers with Hendryk Pfeiffer in week268, and wrote: “Over the complex numbers, or any field of characteristic zero, an algebra is strongly separable iff it’s finite-dimensional and semisimple. The story is trickier over other fields - see that last paper of Lauda and Pfeiffer if you’re interested.”

I must admit that when you were writing this paper, I was not terribly interested in the difference between strongly separable algebras and finite-dimensional semisimple algebras, since this difference only shows up over fields of nonzero characteristic. You see, my years of work with physicists had conditioned me to view all fields other than the complex numbers with a healthy contempt. We may condescend to work with the real numbers at times, but nothing beyond that. I mean, how can you take seriously anyone who wastes time on the possibility that, say, 3=0?

However, with my increasing interest in number theory and algebra in general, I now find these things more attractive! And, it’s especially interesting how you and Hendryk used Frobenius algebras over fields of nonzero characteristic to get new tangle invariants.

Re: This Week’s Finds in Mathematical Physics (Week 268)

for a Frobenius algebra on a Hilbert space, require that the comultiplication is adjoint to the multiplication! We call these ‘†\dagger-Frobenius algebras’…

Later Jamie wrote:

So, for a finite-dimensional Hilbert space, choosing a commutative †\dagger-Frobenius algebra, for which the multiplication is adjoint to the comultiplication, is equivalent to choosing an orthogonal basis.

Am I right in assuming that the phrase ‘for which the multiplication is adjoint to the comultiplication’ is a redundant rhetorical flourish, which could safely be omitted? After all, it seems you’ve just defined a †\dagger-Frobenius algebra to be one where the multiplication is adjoint to the comultiplication.

I don’t mind redundant rhetorical flourishes — I do ‘em all them time. But if I’m confused here, I’d like to be corrected.

Also: if I were defining a †\dagger-Frobenius algebra, I’d also want the unit to be adjoint to the counit. Does this follow from the rest somehow?

Re: This Week’s Finds in Mathematical Physics (Week 268)

Am I right in assuming that the phrase ‘for which the multiplication is adjoint to the comultiplication’ is a redundant rhetorical flourish, which could safely be omitted?

Yes, you’re right!

Also: if I were defining a †-Frobenius algebra, I’d also want the unit to be adjoint to the counit. Does this follow from the rest somehow?

Normally we would assume that †\dagger is strong monoidal, and strong monoidal functors preserve monoids. (Could you even define strong monoidal functors this way, as functors which preserve all monoids? There should be some sort of Strong Microcosm Principle that tells you this would work.) Of course, the †\dagger-functor is contravariant, so it actually turns monoids into comonoids.

But anyway, all the results I’m stating are just designed to work in FdHilb with its usual †\dagger-functor and usual tensor product, and this definitely turns monoids into comonoids.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Thanks for answering my first question, Jamie! I don’t think you actually answered the second one. Since I’m trying to get all the details right, I’ll ask you again:

In a †\dagger-Frobenius monoid, as defined by you and your pals, is the unit the adjoint of the counit?

(I know that †\dagger sends monoids to comonoids, but that’s a different issue.)

Could you even define strong monoidal functors this way, as functors which preserve all monoids?

No: there may be very few monoid objects.

Let’s think of a group as a monoidal category with group elements as objects, multiplication as the tensor product, and only identity morphisms. The only monoid object in here is the identity element!

A strong monoidal functor between groups is then exactly the same thing as a group homomorphism. But a functor that preserves monoid objects is just any function sending the identity of one group to the identity of the other.

Re: This Week’s Finds in Mathematical Physics (Week 268)

John said:

In a †\dagger-Frobenius monoid, as defined by you and your pals, is the unit the adjoint of the counit?

(I know that †\dagger sends monoids to comonoids, but that’s a different issue.)

Since the †\dagger-functor sends monoids to comonoids, then the adjoint of the unit is certainly a counit for the adjoint of the multiplication. But counits for comonoids are unique, just like units for monoids! So if a comonoid has comultiplication given by the adjoint of a monoid’s multiplication, it must also have counit given by the adjoint of the monoid’s unit.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Jamie wrote:

If you really want it succinctly…

I do!

yes!

Great!

But also thanks for pointing out that this follows from the rest. For some reason I’d never noticed that the usual proof that the unit of a monoid is unique generalizes painlessly to a monoid object in any monoidal category!

Re: This Week’s Finds in Mathematical Physics (Week 268)

Jamie Vicary’s reasonable desire to use ‘special’ as the name for Frobenius algebras where comultiplication followed by multiplication is the identity. I hope everyone else in the universe follows suit!

Jamie’s correction of my false claim that
“taking a Hilbert space and making it into a commutative separable Frobenius algebra is the same as equipping it with an orthonormal basis”.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Now I’ve adjusted week268 in an attempt to fix the problem Todd pointed out, and give a definition of *-autonomous category that’s equivalent to the usual one. I got a lot of help from Paul–André Melliès, but of course I’m to blame for any remaining errors:

But now, I want to give a definition of *-autonomous categories,
which simultaneously makes it clear that they’re natural structures
in logic, and that they’re categorified Frobenius algebras!

Suppose A is any category. We’ll call its objects
“propositions” and its morphisms “proofs”.
So, a morphism

f: a → b

is a proof that a implies b.

Next, suppose A is a symmetric monoidal category and call the tensor
product “or”. So, for example, given proofs

f: a → b, f’: a’ → b’

we get a proof

f or f’: a or a’ → b or b’

Next, suppose we make the opposite category Aop into a
symmetric monoidal category, but with a completely different tensor
product, that we’ll call “and”. And suppose we have a
monoidal functor:

not: A → Aop

So, for example, we have

not(a or b) = not(a) and not(b)

or at least they’re isomorphic, so there are proofs going both ways.

Now we can apply “op” and get another functor I’ll also call
“not”:

not: Aop → A

Using the same name for this new functor could be confusing, but it
shouldn’t be. It does the same thing to objects and morphisms; we’re
just thinking about the morphisms as going backwards.

Next, let’s demand that this new functor be monoidal! This too is
quite reasonable; for example it implies that

not(a and b) = not(a) or not(b)

or at least they’re isomorphic.

Next, let’s demand that this pair of functors:

not
--------->
A A^{op}
<----------
not

be a monoidal adjoint equivalence. So, for example, there’s a
one-to-one correspondence between proofs

not(a) → b

and proofs

not(b) → a

Now for the really fun part.
Let’s define a kind of “bilinear form”:

g: A × A → Set

where g(a,b) is the set of proofs

not(a) → b

And let’s demand that g satisfy the Frobenius
axiom! In other words, let’s suppose there’s a natural
isomorphism:

g(a or b, c) ≅ g(a, b or c)

Then A is a “*-autonomous category”! And this is
a sensible notion, since it amounts to requiring a natural
one-to-one correspondence between proofs

In case it slipped by too fast, let me repeat the definition of
*-autonomous category I just gave. It’s a symmetric monoidal
category A with a monoidal adjoint
equivalence called “not” from A (with one tensor product,
called “or”) to Aop (with another, called
“and”), such that the functor

g: A × A → Set
(a,b) |→ hom(not(a),b)

is equipped with a natural isomorphism

g(a or b, c) ≅ g(a, b or c)

I hope I didn’t screw up. I want this definition to
be equivalent to the usual one,
which was invented by Michael Barr quite a while ago:

Namely, they show a *-autonomous category is the same as a symmetric monoidal category A equipped with a contravariant adjoint equivalence

not: A → A

which is equipped with a “strength”, and where the unit and counit of the adjunction respect this strength.

Later, in his paper “Frobenius monoids and pseudomonads”, Street
showed that *-autonomous categories are precisely Frobenius
pseudomonoids in a certain monoidal bicategory with:

categories as objects;

profunctors (also known as distributors) as morphisms;

natural transformations as 2-morphisms.

Alas, I’m too tired to explain this now! It’s a slicker way of saying what I already said. But the cool part is
that this bicategory is like a categorified version of Vect, with
the category of finite sets replacing the complex numbers. That’s
why in logic, the “nondegenerate bilinear form” looks like

Re: This Week’s Finds in Mathematical Physics (Week 268)

(My emphasis.) As I understand it – and I’d love to be corrected if I’m wrong – Frobenius pseudomonoids in Prof are actually a little more general than star-autonomous categories. Certainly every star-autonomous category gives rise naturally to a Frobenius pseudomonoid in Prof, but I doubt that every Frobenius pseudomonoid arises in this way.

The converse probably does hold when the underlying category is Cauchy-complete, i.e. all idempotents split. At least I think it does, though I haven’t carefully checked. And of course every category is Morita equivalent to a Cauchy complete one, so it’s perfectly plausible that the two notions are indeed equivalent in some precise sense.

I imagine it could be shown that the bicategory of Frobenius pseudomonoids in Prof is biequivalent to the bicategory of star-autonomous categories, where both these bicategories are defined in some reasonable way. Perhaps that is what you meant all along, in which case I withdraw my objection. :-)

Re: This Week’s Finds in Mathematical Physics (Week 268)

I will need to make some more corrections to handle Robin and Jamie’s objections. I forgot about the requirement for Cauchy completeness, and I guess I’ll need to check to see if anything forces the relevant profunctors to actually be functors — I don’t see any such thing.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Further, we have “Frobenius pseudomonoids”, categorified Frobenius algebras inside monoidal bicategories, and are given an example of one inside a symmetric monoidal bicategory. So where are the very nice braided monoidal bicategories, which categorify modular tensor categories? And what kind of categorified conformal field theory might be expected there?

Re: This Week’s Finds in Mathematical Physics (Week 268)

Hom(not(a),b)Hom(not(a), b) is the same as Hom(not(a)andnot(b),false)Hom(not(a) and not(b), false), or Hom(not(aorb),false)Hom(not(a or b), false).

So, g(a,b)≅g(aorb,false)g(a, b) \cong g(a or b, false), the monoidal product in AA followed by a cup process which looks for proofs of absurdity.

How do caps and cups work here? The empty proposition in AA with ‘or’ as product is ‘false’. For any proposition there’s a birth/cap process which is a proof of it from false (ex falso quodlibet).

But don’t we need a death/cup process for each proposition for the Frobenius structure?

Maybe I’m getting confused because in the TQFT case you treat earlier, a diagram is a specific map, e.g., the cup for a vector space is a specific map to the complex numbers. In the propositional case you seem to want it to be a (possibly empty) set. But then do all other diagrams represent sets of proofs?

Re: This Week’s Finds in Mathematical Physics (Week 268)

So, g(a,b)≅g(aorb,false)g(a,b) \cong g(a or b,false), the monoidal product in AA followed by a cup process which looks for proofs of absurdity.

Right — but let’s not call it a ‘cup process’, since people often use ‘cap’ to stand for the copairing:

and ‘cup’ to stand for the pairing:

while you are using ‘cup process’ to mean the ‘counit’:

Of course, your cup process is an honest cup-shaped cup! But, I was still confused by what you wrote for a while.

Terminology aside, you’re saying that the pairing is multiplication followed by the counit, as this equation indicates:

To get this pairing to admit a copairing that satisfies the zig-zag equation, I think we need to replace our functors by profunctors, also known as distributors. A profunctor from a category CC to a category DD is a functor C×Dop→SetC \times D^{op} \to Set. I was trying hard to say as much as I could about the use of categorified Frobenius algebras in logic without getting to the point where we need profunctors! You’re attempting to push me over the brink, into the sea of profunctors. So, you should read Street’s paper.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Re: This Week’s Finds in Mathematical Physics (Week 268)

To really understand all these cups and caps you need to think about 2-manifolds as 2-morphisms in a symmetric monoidal 2-category with duals at all levels.

This counit:

descibes the disappearance of a 1-sphere, or circle. But this process is really the cancellation of an upper semicircle and its dual, the lower semicircle. A counit describes the cancellation of something and its dual.

But the lower semicircle is itself a counit, describing the disappearance of a 0-sphere — that is, the cancellation of the positively oriented point and its dual, the negatively oriented point!

The ‘pairing’ is also a counit:

describing the cancellation of a circle and its dual, an oppositely oriented circle.

But, since the circle is itself the composite of the upper semicircle and the lower semicircle, the pairing can be factored into two separate process, as shown at right. First we do multiplication, which cancels two semicircles:

Re: This Week’s Finds in Mathematical Physics (Week 268)

Regarding the pair of pants as cancelling semicircles brings to mind the Zwiebach-HIKO description of closed string field theory in which 2 closed strings (maps of the circle to a target space where the circle is the standard and parameterized by theta) interact fuse if a semicircle of one is equal to a semicircle of the other except with reverse parameterization. Or a single closed string can split into many via ANY
arc from the image of theta to the image of theta + pi.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Yes! That’s what all that ‘profunctor’ baloney is about. I’m sorry my previous reply to your questions was so obscurantist.

Let’s consider the humble counit as an example:

In logic, it corresponds to a profunctor

e:A→1 e: A \to 1

where AA is our *-autonomous category. This is a fancy way of saying it’s a functor from AA to SetSet. Objects of AA are propositions, and this functor assigns to each proposition aa the set of proofs leading from

not(a)not(a)

to

falsefalse

In other words,

e(a)=hom(not(a),false)e(a) = hom(not(a),false)

You seem to have almost guessed this. I think a less negative way of putting it should be

e(a)=hom(true,a)e(a) = hom(true,a)

Since we’re not doing anything intuitionistic, this should be the same.

MORAL: the counit encodes the logical constant ‘true’.

Similarly, the comultiplication

corresponds to a profunctor

m:A→A⊗Am : A \to A \otimes A

which is another name for a functor

m:A×Aop×Aop→Setm : A \times A^{op} \times A^{op} \to Set

and this functor is:

m(a,b,c)=hom(bandc,a)m(a,b,c) = hom(b and c, a)

Note this functor is covariant in aa and contravariant in the other two variables, as it should be.

MORAL: the comultiplication encodes the logical operation ‘and’.

EXERCISE: work out the profunctors corresponding to unit and multiplication.

HINT: they must be ‘upside down versions’ of what we’ve done.

EXERCISE: guess how to compose profunctors and check that some of the Frobenius axioms hold (up to isomorphism). Also check that

g(a,b)≅g(aorb,false)g(a,b) \cong g(a or b, false)

as I claimed.

HINT: Profunctors are like categorified tensors, with the variables a,b,c,a,b,c, as indices. So, composing profunctors is a categorified version of ‘summing over repeated indices’.

Re: This Week’s Finds in Mathematical Physics (Week 268)

The unit is profunctor from 1 to AA, so corresponds to a functor from AopA^{op} to Set. Looks like it sends aa to Hom(true,not(a))Hom(true, not(a)) or Hom(a,false)Hom(a, false).

Multiplication is a profunctor from A×AA \times A to AA, or a functor from A×A×AopA \times A \times A^{op} to Set. Looks like (a,b,c)(a, b, c) is sent to Hom(c,aorb)Hom(c, a or b).

One unit law relates the sum over aa of Hom(c,aorb)×Hom(a,false)Hom(c, a or b) \times Hom(a, false) with Hom(c,b)Hom(c, b).

Hmmm. A pair from the product gives a proof of bb from cc, but are we expecting/achieving isomorphisms? Oh, isn’t there something about degeneracy in classical logic, all proofs of bb from cc are the ‘same’?

Re: This Week’s Finds in Mathematical Physics (Week 268)

The unit is a profunctor from 1 to AA, so corresponds to a functor from AopA^{op} to Set. Looks like it sends aa to Hom(true,not(a))Hom(true, not(a)) or Hom(a,false)Hom(a, false).

Multiplication is a profunctor from A×AA \times A to AA, or a functor from A×A×AopA \times A \times A^{op} to Set. Looks like (a,b,c)(a, b, c) is sent to Hom(c,aorb)Hom(c, a or b).

Right!

So, multiplication is

hom(b and c,a)

while comultiplication is

hom(a, b or c).

The unit is

hom(a,false)

while the counit is

hom(true,a).

Ain’t it cool how flipping our cobordisms upside down — in physics, reversing the arrow of time — corresponds in logic to simultaneously switching ‘true’ and ‘false’, ‘and’ and ‘or’, and turning around the direction of implication?

It’s stuff like this that makes me think there really is just one kind of duality, with tentacles reaching in all directions.

One unit law relates the sum over aa of Hom(c,aorb)×Hom(a,false)Hom(c, a or b) \times Hom(a, false) with Hom(c,b)Hom(c, b).

Hmmm. A pair from the product gives a proof of bb from cc, but are we expecting/achieving isomorphisms? Oh, isn’t there something about degeneracy in classical logic, all proofs of bb from cc are the ‘same’?

That would amount to saying our *-autonomous category AA is equivalent to a mere poset. I sure hope things work generally than that! I hope the problem is that you’re ‘summing over indices’ in a slightly naive way: taking a disjoint union as aa varies, instead of a more clever colimit.

In other words, I think we need to consider the morphisms between various choices of the index aa. If we had zillions of isomorphic propositions aa, your sum over aa would be enormous, but having all those isomorphic copies shouldn’t change the answer at all. When there are morphisms that aren’t isomorphisms it’s subtler, but we still want to mod out by ‘redundancies’ somehow. That means doing some sort of colimit: taking a disjoint union but then modding out.

Physicists sum over repeated indices —but only when one index appears ‘upstairs’ and the other ‘downstairs’. Here we are summing over repeated objects like aa — but only when one appears covariantly and the other contravariantly. Hmm, that sounds awfully similar!

As an explanation of coends, this Wikipedia article is useless: you’ll find it much easier to figure out the right way to compose profunctors yourself, without knowing or caring about coends. The article will only intimidate you and make the process slower… except for the notation they use for coends, which looks temptingly like ‘summing over repeated indices’.

Re: This Week’s Finds in Mathematical Physics (Week 268)

David wrote:

Oh, isn’t there something about degeneracy in classical logic, all proofs of bb from cc are the ‘same’?

Since a kindly elf helped me access Brady and Trimble’s paper, I found that they give a precise statement of this fact, which they attribute to Joyal: any *-autonomous category in which andand is the cartesian product is necessarily a poset — so, any two morphisms from cc to bb are equal. In fact, it’s a Boolean algebra!

Re: This Week’s Finds in Mathematical Physics (Week 268)

To check zigzag identies we could do with the copairing cap, the composite of comultiplication and unit. That would seem to be h(a,b)≅hom(aandb,false)≅hom(b,not(a))h(a, b) \cong hom(a and b, false) \cong hom(b, not(a)).

Re: This Week’s Finds in Mathematical Physics (Week 268)

Yes, I agree that’s how it must work. It’s still interesting to figure out precisely how we’re composing profunctors when doing this sort of computation. The stuff about coends sounds scary at first, but I think it boils down to something quite intuitive. For example, in the calculation you just did, I think you’re secretly forming the sets

hom(not(c),b)×hom(b,not(a))hom(not(c),b) \times hom(b, not(a))

and then taking a colimit over choices of the intermediate proposition bb to get the set

hom(not(c),not(a))hom(not(c), not(a))

This should amount to saying that every proof that not(c)not(c) implies not(a)not(a) factors through some intermediate proposition bb… but in a highly non-unique way! So, the set of proofs

Re: This Week’s Finds in Mathematical Physics (Week 268)

We already got confused about this here! It’s all still conjectural, I think, but the consensus is that 3Cob2{}_2 should be the free strongly symmetric monoidal 2-category on a braided Frobenius pseudomonoid, which has multiplication left- and right-adjoint to its comultiplication, and unit left- and right-adjoint to its counit.

As a reminder, 3Cob2{}_2 is the 2-category with finite collections of oriented circles as objects, compact 2-manifolds with boundary as morphisms, and compact 3-manifolds with boundary and corners as 2-morphisms.

If I’ve got this a bit wrong, I’m sure Bruce or John will correct me pretty quickly!

Re: This Week’s Finds in Mathematical Physics (Week 268)

??? IS THE FREE SYMMETRIC MONOIDAL BICATEGORY ON A SYMMETRIC FROBENIUS PSEUDOMONOID.

This is a very interesting puzzle!

(Street uses ‘Frobenius pseudomonoid’ instead of ‘categorified Frobenius algebra’; since he invented the definition perhaps we should use his terminology.)

If you take the free symmetric monoidal bicategory on a braided Frobenius pseudomonoid with some extra bells and whistles related to duality, you should get 3Cob23Cob_2 — as Jamie notes, I wrote about this
before.
I’m not sure Jamie correctly listed all the bells and whistles related to duality; Aaron Lauda has a partially written paper on this conjecture which tackles certain nuances that I’m reluctant to delve into here.

But the question you raise should have an answer that’s not particularly related to 3d topology… since you’re talking about a symmetric rather than braided Frobenius pseudomonoid.

Re: This Week’s Finds in Mathematical Physics (Week 268)

I was wondering whether if we could find this free bicategory, since your formulation of propositional calculus took the form of a symmetric Frobenius pseudomonoid, we could understand and possibly improve diagrammatic representations of the calculus, such as Peirce’s alpha graphs.

Is there a special name given to pseudomonoids in the bicategory of categories and profunctors?

Re: This Week’s Finds in Mathematical Physics (Week 268)

If anyone could help us understand the significance for logic of your proposed “free symmetric monoidal bicategory on a symmetric Frobenius pseudomonoid”, I imagine it would be Todd Trimble. Indeed I want to reread his paper on the work of Peirce, since I get the feeling I’m slowly and feebly rediscovering things he talked about there.

Unfortunately, right now I can only access this paper if I feed an evil giant. I wish Todd or Gerry Brady would make this paper freely available online (hint hint).

Is there a special name given to pseudomonoids in the bicategory of categories and profunctors?

‘Promonoidal categories’. Just one more part of Australia’s bid for global dominance.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Re: This Week’s Finds in Mathematical Physics (Week 268)

David wrote:

Do you feel you answered Todd’s suspicion?

Yes! Instead of leaving the categorified Frobenius axiom

g(a or b, c) ≅ g(a, b or c)

for last and idiotically hoping it would follow from the rest, my corrected exposition includes it as an axiom. This was supposed to be the point all along: the Frobenius axiom says something interesting about logic!

But this raises tons of puzzles, like: can you use a system of propositional logic — or more precisely, a *-autonomous category — to define some sort of categorified version of a 2d TQFT? Apparently so! But how does it work, technically? Do we get some sort of weak 2-functor

Z:2Cob→Cat?Z : 2Cob \to Cat ?

What kind, exactly? And regardless of the technical details, what does it all mean???

I have a vague memory of Todd trying to explain all this stuff to me a long time ago, in Cambridge.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Tillmann’s Frobenius categories[54] Tillmann suggests that in order to encode 3-dimensional information into a 2-dimensional topological quantum field theory one must consider a more interesting version of the 2-dimensional cobordism category, namely 2Cob22 Cob_2. The objects of 2Cob22 Cob_2 are closed, oriented, compact 1-manifolds, and the morphisms are oriented, compact 2-manifolds. The 2-morphisms of this 2-category are the connected components of orientation preserving diffeomorphisms of the 2-manifolds. This cobordism 2-category was first studied by Carmody [11].

By extending the category 2Cob2 Cob to the 2-category 2Cob22 Cob_2, Tillmann defines a modular functor as a monoidal 2-functor from 2Cob2→k−Cat2 Cob_2 \to k-Cat, where k−Catk-Cat is the 2-category of linear categories, linear functors and linear natural transformations. Tillmann calls the image of such a 2-functor a ‘Frobenius category’. She goes on to show that these Frobenius categories are related to 3-dimensional topological quantum field theory. In our terminology, these
‘Frobenius categories’ are a symmetric version of pseudo Frobenius algebras in the 2-category of kk-linear categories.

This example is particularly related to the results of this paper. In the next section we will discuss the nonsymmetric version of the cobordism 2-category described above. The effect of removing the symmetry requirement amounts to ‘smashing the cobordisms flat’ into what we call 3Thick3 Thick, the 2-category of 3-dimensional thick tangles.

So, does 2Cob22 Cob_2 provide the best geometric rendition of propositional logic? And where do your fake surfaces feature?

Re: This Week’s Finds in Mathematical Physics (Week 268)

I have a query on referencing.

In each dagger special Frobenius algebra in a dagger symmetric monoidal category each morphism composed from the dagger symmetric monoidal structure and algebra structure of which the diagrammatic representation is connected is completely determined by its number of inputs and outputs. (if it is not special then an additional piece of data is the number of ‘holes’ cf 1.4.16 in Joachim Kock’s book for the case of 2-Cob) We use this fact all over the place since it gives an amazingly simple diagrammatic way to reason e.g quant-ph/0608072 and also generalisations of it in 0808.1029 and in several forthcoming papers.

We sketched a proof of it in quant-ph/0608072 and typically refer to it as ‘spider thm’ (due to the resulting graphical normal form) but surely this must have been known in quantum algebraic circles as well as in abstract category theory.

To whom should this result be attributed (or is it some piece of folklore) and where has it previously been used for diagrammatic reasoning outside of the context of TQFT?

Re: This Week’s Finds in Mathematical Physics (Week 268)

Bob wrote:

In each dagger special Frobenius algebra in a dagger symmetric monoidal category each morphism composed from the dagger symmetric monoidal structure and algebra structure of which the diagrammatic representation is connected is completely determined by its number of inputs and outputs.

I don’t think we even need the dagger business for a result very similar to this. So, I think it’s best to derive what you need from known results on commutative special Frobenius algebras.

To whom should this result be attributed (or is it some piece of folklore) and where has it previously been used for diagrammatic reasoning outside of the context of TQFT?

In Section 5.4 he shows that the PROP for ‘commutative separable algebras’ is the same as the category of cospans of finite sets.

He could equally well have said the PROP for ‘commutative special Frobenius algebras’ is the same as the category of cospans of finite sets.

Let me translate his result into less technical language, so it’s clear it gives what you want!

Suppose you have any commutative special Frobenius algebra AA in any symmetric monoidal category. Then, to specify an operation A⊗m→A⊗nA^{\otimes m} \to A^{\otimes n}, it suffices to specify functions

m→fk←gn m \stackrel{f}{\rightarrow} k
\stackrel{g}{\leftarrow} n

where I’m thinking of n,mn,m and kk as finite sets with those numbers of elements.

The idea is that this pair of functions:

m→fk←gn m \stackrel{f}{\rightarrow} k
\stackrel{g}{\leftarrow} n

describes a string diagram with mm inputs, nn outputs and kk connected components. The functions ff and gg say how the input and output edges sit inside the set of connected components.

(Since our Frobenius algebra is special, we don’t need to specify how many ‘holes’ each component has!)

This gadget:

m→fk←gn m \stackrel{f}{\rightarrow} k
\stackrel{g}{\leftarrow} n

is called a ‘cospan of finite sets’. There’s a well-known way of composing cospans, which is the same as composing string diagrams and then using the ‘specialness’ condition to kill off any holes that get formed. Lack’s result includes the fact that this way of composing cospans is how you compose operations when you’ve got any commutative special Frobenius algebra.

I discussed this result in week268. I initially failed to credit Steve Lack for this result, since it also appears in a later paper by someone else.

Here’s what the current version says:

If we use a commutative special Frobenius algebra to get a 2d TQFT,
it fails to detect handles! That seems sad. But these papers:

Cospan(FinSet) IS THE
FREE SYMMETRIC MONOIDAL CATEGORY
ON A COMMUTATIVE SPECIAL FROBENIUS ALGEBRA.

Here Cospan(FinSet) is the category of "cospans" of finite sets.
The objects are finite sets, and a morphism from X to Y looks like
this:

X Y
\ /
F\ /G
\ /
v v
S

If you remember the "Tale of Groupoidication" starting in
If you remember the "Tale of Groupoidication" starting in
"week247", you’ll know about
spans and how to compose spans using pullback. This is just the same
only backwards: we compose cospans using pushout.

But here’s the point. A 2d cobordism is itself a kind of cospan:

X Y
\ /
F\ /G
\ /
v v
S

with two collections of circles included in the 2d manifold S. If we
take connected components, we get a cospan of finite sets. Now we’ve
lost all information about handles! And the circle - which was a
commutative Frobenius algebra - becomes a mere one-point set - which
is a special commutative Frobenius algebra.

Re: This Week’s Finds in Mathematical Physics (Week 268)

John, thanks for explaining this! I am aware that Steve Lack, Ross Street and other Australians are doing amazing stuff in this area but it’s just to far beyond what a half-baked pseudo-categoretician like me can grasp.

Re: This Week’s Finds in Mathematical Physics (Week 268)

The once-intractable problem “is there such a thing as a Boolean category” - a category which is to a Boolean algebra what a cartesian-closed category with coproducts is to a Heyting algebra - has been given several solutions during the last six years or so. But, we still cannot say we have a really satisfying solution.

The present work started as a denotational semantics for classical logic in the category of posets and bimodules, furthering the work in [Lam07]. After noticing that the objects representing Boolean propositions were equipped with a Frobenius algebra structure, it was also found that one could use it to get a faithful representation of the “Free Frobenius category”, i.e. the free symmetric monoidal category generated by one object equipped with a Frobenius algebra structure [Dij89]. Actually one can also get a faithful representation of the “Free Frobenius compact-closed category”, which happens to be a minor variation on it. In addition, with a little bit more work, we managed to construct subcategories of Posets and bimodules which are equivalent to these two free monoidal categories, thus giving a purely semantical construction for these syntactical/geometric objects.

The cleanest way to represent a category of logical formulas and proofs is by the means of so-called proof nets, where the objects are ordinary formulas but the maps belong to a category C where some logically distinct objects have been identified (e.g., conjunction and disjunction). The original structure of the logical formulas introduces constraints on these maps, so there is a faithful functor from the proof net category to C, but one which is neither full nor injective on objects.

The work above allows us to construct a new category of proof nets for classical logic where the category C is a free Frobenius category, and give the required “full completeness theorem” i.e., show that every map in that category comes from a proof in the sequent calculus. This use of Frobenius algebras in classical logic is quite different from the one proposed by Hyland [Hyl04] and furthered by Garner [Gar05].

It more resembles the work of Lamarche-Strassburger [LS05], where the category C is built from sets and relations using an “interaction category” construction [Hyl04, Section 3], where composition is obtained by the means of a trace operator. Our new category has the desirable property of being resource-sensitive, i.e. counting how many times contractions have been used to superpose bits and pieces of a proof.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Re: This Week’s Finds in Mathematical Physics (Week 268)

I’m new to this topics and I’m trying to understand the notions of Frobenius pseudomonoids defined by Street in “Frobenius monads and pseudomonoids”.

At the introduction of this paper he says: “Some connection between quantum groups and Frobenius algebras is already apparent from the fact that quantum groups are Hopf algebras and finite-dimensional Hopf algebras are Frobenius [LSw]. We intend to deepen the connection between Frobenius algebras and quantum group theory.”.

I don’t really know that much about the connection between Frobenius algebras and quantum groups to be honest (except the one he mentioned). Where can I find a good summary of the most connections between these two topics? And how do the Frobenius pseudomonoids generalize these “already known” connections? (a good reference or explanation would be really helpfull).

I already tried to search on the internet but I’m always running into the same articles: “Frobenius monads and pseudomonoids” of Street and “Frobenius algebras and ambidextrous adjunctions” of Lauda. I’ve tried to read them but they don’t really explain a connection between quantum groups and frobenius algebras or how they abstract that notion.

Re: This Week’s Finds in Mathematical Physics (Week 268)

Aaron Lauda was my student back when he was getting a masters in physics at UCR. Those were some of his first papers, and I still like them a lot.

But maybe you need to think about this: if GG is a finite group, the group algebra ℂ[G]\mathbb{C}[G] is both a Hopf algebra and a Frobenius algebra. The multiplication is the same in both cases, but the comultiplication is different. For the Hopf algebra it’s

All the fancier relations between Hopf algebras and Frobenius algebras, and their categorified versions, are based on this idea. So if you’re not comfortable with this, you should start here.

I don’t know a good reference, except for the standard references on Hopf algebras and Frobenius algebras. There are some papers on Hopf–Frobenius algebras, which discuss the interaction between the Hopf and Frobenius coproducts, but I don’t know one I like: they’re easy to find using Google.